Effects of density differences on lateral mixing in open-channel flows

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EFFECTS OF DENSITY DIFFEHENCES ON 'LATERAL MIXING IN OPEN-CHANNEL FLOWS

Thesis by

Edmund Andrew Prych

In Partial Fulfillment of the Requiren1ents ·For the Degree of

Doctor of

Philosophy-California Institute of Technology ·Pasadena, California

1970

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-ii

-/\t :1' NOWLED< i.EM ENTS

This st:udy, one of a group of invcsti gatious al. the Californi ;t Institute uf Technology titled "Dispersion in Hydrologic and Coastal Environments", was funded by the Federal w·ater Pollution Control Administration through Grants No. 16000 DGY and No. 16070 DGY. During the study the writer also received financial assistance in the form of stipends and tuition payments while a U.S. Public Health .Scrv.i ce 'I' r ai nee (l 96 7 -70) and while a National Science Foundation Trainee (19(16-67). The writer thanks each of these agencies for their s11 pport.

To Dr. Norrnan H. Brooks, adviser and fruitful source of 11_the right questions 11

, the writer expresses his sincere gratitude. The writer also thanks Dr. Vito A. Vanoni for comments made during the study, and Dr. E. John List for his comments during the writing of

this thesis.

For their assistance in constructing and modifying laboratory equipn1ent, the writer offers a hearty thanks to Mr. Elton F. Daly, s11pc'rviso1· of the shop and laboratory; to Mr. Robert L. Greenway, his ta]enlt·d assistant; and to Mr. Carl A. Green, Jr. , who also pre

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-111-performed a variety of tasks during this investigation; they are: l'vfcssrs. Raul Basu, George Chan, Yoshiaki Daimon, Edward F.

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-iv-ABSTRACT

This study investigates lateral rnixing of tracer fluids in

turbulent open-channel flows when the tracer and an1bicnt fluids have different densities. Longitudinal dispersion in flowo with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density

gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate,

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-v-to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreernent with the data. However, predicted longitudinal salinity distributions in an

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Chapter

1.

2.

3.

V l

-TABLE OF CONTENTS

INTRODUCTION 1. I Purpose

1. 2 Method of Investigation

BACKGROUND AND REVIEW OF PREVIOUS WORK 2. 1 The Conservation Equation

2. 2 Mass Transport in Open-Channel Flows of Homogeneous Density

2. 2. 1 Longitudinal Dispersion 2. 2. 2 Longitudinal Diffusion 2. 2. 3 Lateral Diffusion

1

I 2

5

6

6 8 10 2. 2. 3. 1 Diffusion at the Free Surface 10 2. 2. 3. 2 Depth-Averaged Diffusion

Coefficient 10

2. 2. 4 Vertical Diffusion 15

2. 3 Effects of Density Differences on Mass Transfer 16 2. 3. 1 Vertical Density Gradients 16 2. 3. 2 Horizontal Density Gradients 17

EXPERIMENTAL APPARATUS AND TECHNIQUES 20

3. 1 Description of Experiments 20

3. 2 Flume 21

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Chapter

4.

3. 3

3. 4

-vii-TABLE OF CONTENTS (Cont'd)

3. 2. 2 Water Circulation 3. 2. 3 Rough Bottom

Watc1· D<·pth and Velocity McaH11ring Eq11ip:r11enf: Tracer Fluid System

3. 4. 1 Fluid Handling System 3.4.2 Sources

21

24

?,(, 26

26

28

3.4.2. 1 Narrow Sources 28

3. 4. 2. 2 The Intermediate Width Sources 31

3. 4. 2. 3 Wide Source 33

3. 5 Fluid Sampling Apparatus 3. 6 Sample Analysis

3. 7 3. 8

3.9

3. 6. 1 Description of the Technique 3. 6. 2 Accuracy of the Technique Measurements of Fluid Density Experimental Procedures

3. 8. 1 3. 8. 2

3. 8. 3

Preparation

Positioning the Sampling Tubes Taking the Samples

Experiments With Floating Particles 3. 9. 1 Apparatus

3.9.2 Procedures

EXPERIMENT AL DAT A 4. 1 Hydraulic Data

35 38 38 42 43 45

45

46

47 49 49 51

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-viii-TABLE OF CONTENTS (Cont'd)

Chapter

4. 1. 1 Flow Condition Code and Flume Bottom 5 4

4. 1. 2 Water Depth S4

4. 1. 3 Slope 56

4. 1. 4 Shear Velocity 58

4. 1. 5 Metered Discharge and Discharge

Velocity 58

4. 1. 6 Velocity in Central 60 cm of Flume 58

4. 1. 7 von Karman1_{s k} ₅₉

4. 1. 8 Friction Factors 62.

4. 1. 9 Reynolds Number and Froude Number 62 4. 2 Experiments With Fluid Tracers 63

4. 2. 1 Introduction 63

4. 2. 2 Concentration Distributions in

Cross-Sections 65

4. 2. 3 Depth-Averaged Concentration

Distributions 79

4. 2. 3. 1 Calculating the

Depth-Averaged Concentrations 79 4. 2. 3. 2 The Lateral Distributions of

C

80

4. 2. 4 Variances of the Lateral Concentration

Distributions 95

4. 2. 4. 1 Calculations of the Variances 95

4. 2. 4. 2 Presentation of the Variances 103

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Chapter

5.

6.

-ix-TABLE OF CONTENTS (Cont'd)

4. 3 Experiments With Floating Particles 4. 3. 1 Calculating the Variance

4. 3. 2 Presentation of Data

DAT A ANALYSIS

Page

135

140

145

5. 1 Experiments Without Density Differences 145 5. 1. 1 Experiments With Tracer Fluids 145

5. 1. 2 Experiments With Floating Particles 147 5. 2 Experiments With Density Differences 149

5. 2. 1 Introduction 149

5. 2. 2 Selection of Dimensionless Numbers 149 5. 2. 3 The Dimensionless Variance 152

5.2.3. 1 The Excess Variance 5. 2. 3. 2 The Fraction r

153

160

5. 2. 3. 3 Maximizing Lateral Mixing 163 5. 2. 4 The Average Coefficient of Variation 166

5. 2. 5 Predicting Concentration Distributions 168

ANALYTIC STUDY 6. 1 Introduction 6. 2 Basic Equations

6. 3, Simplification of the Basic Equations 6. 3. 1 Lateral Mixing

6. 3. 2 Longitudinal Mixing

170

173

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-x-TABLE OF CONTENTS (Cont'd) Chapter

6. 4 Solution for the Velocity w and the Dispersion

Coefficient Dz 18 1

6. 5 Application of Theory to Experiments of This

Study 186

6. 5. 1 Sources of Finite Width 186 6. 5. 2 Two Wide Parallel Streams 191 6. 6 Application of Theory to MIT Experiments 193 6. 6. 1 Description of Experiments 193

6. 6. 2 Theoretical Analysis of Experiment 196 6. 6. 3 Comparison of Theory With Experiment 198

7. SUMMARY OF RESULTS 204

7. 1 Experiments 204

7. 1. 1 Experiments Without Density 204 Differences

7. 1. 2 Experiments With Density Differences 205

7. 2 Theory 207

LIST OF SYMBOLS 209

LIST OF REFEHENCES 214

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Nun1ber

") 1

3. 1

3. 2

3.4

3. 5 3. 6 3. 7

3.

8

3.

9

3. l 0

3. 11

3. 12

4. 1

4.2

-·

Xl-LIST OF FIGUHES

Description

Longitudinal pressure gradients and velocity distributions in an estuary

The 40-n1cter flun1e

Photograph of expanded metal lath used for

roughness on the bottom of the flume

Dirnensioned sketch of metal lath used for roughness on the bottom of the flume

Schematic drawing of the tracer-fluid handling system

The 1-cm-wide source

The 20-cm-wide source

Photograph looking down the flume The m.anifold for the wide source

A fluid sampling rake

The pressure box and sampling rakes on the instrument carriage

Sample of polyethlene particles used in study of

lateral diffusion at the free surface

Compartmented-sieve particle collector; the compartments are 1 cm wide

Profiles of the energy grade line, water surface,

and flume bottom for flow S2

Cross -sections of the flume showing isovels 25 meters downstream from the flume inlet

Pa~c

·---""'-'·

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-xii-LIST OF FIGUHES (Cont'd)

Nun1ber Description Page

4. 3 Velocity profiles observed 25 meters downstream

from the flume inlet 6 l

4. 4 Lines of equal relative concentration in

cross-sections downstream from a 1-cm-wide source

which discharged a fluid with a density the same as the

ambient fluid and with a relative concentration of 1. 0. 67

cross-sections downstream from a I-cm-wide source

which discharged a fluid with a density 0. 0160 gm/cm3

greater than the ambient fluid and with a relative

concentration of l. 0 68

cross-sections downstream from a I-cm-wide source

4. 7

4. 8

4

.

9

4. 10

4. 11

4. 12

which discharged a fluid with a density O. 0158 gm/cm3

less than the ambient fluid and with a relative

concentration of 1. 0

Cross-section of flume showing density-induced secondary circulation patterns downstream from a source of narrow or intermediate width

Cross -sections with lines of equal relative

concentration downstrea:rn from the confluence

of two wide parallel strea:rns of the same density

Cross-sections with lines of equal relative

concentration downstream from the confluence

of two wide parallel streams of different density

Cross-sections of the flume showing dens

ity-induced secondary circulation pattern downstream

from the confluence of two wide parallel streams

of different density

Definition sketch for terms in Eq. 4. lb

Depth-averaged concentration distributions

down-stream from a I-cm-wide source which discharged

a fluid with a density the same as the ambient fluid

and with a relative concentration of 1. O; Exp. 125

69

71

76

77

78

80

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Number 4. 13

4. I4

4. I5

4. I6

4. 17

4. I8

4. 19

4.20

4. 2 I

4.22

-xiii--LIST OF FIGURES {Cont'd)

.Description Page

Depth-averaged concentration distributions down-stream from a I-cm-wide source which discharged a fluid with a density the same as the ambient fluid

and with a relative concentration of 1. O; Exp. l I6 83 Depth-averaged concentration distributions down

-stream from a I-cm-wide source which discharged a fluid with a density 0. 0160 gm/ cm3 more dense than the ambient fluid and with a relative concen

-tration of 1. O; Exp. I I3 84

Depth-averaged concentration distributions dow n-stream from a I-cm-wide source which dischar ged a fluid with a density 0. 0158 gm/cm3 less than the ambient fluid and with a relative concentration of

1. 0; Exp. 128 85

Depth-averaged concentration distributions d own-stream from a 20-cm-wide source which discharged a fluid with a density the same as the ambient fluid

and with a relative concentration of 0. 0370; Exp. 150 88 Depth-averaged concentration distributions

down-stream from a 20-cm-wide source which discharged a fluid with a density O. 00186 gm/ cm3 more dense than the ambient fluid and with a relative concen

-tration of O. 0385; Exp. 152 89

Depth-averaged concentration distributions down -stream from the confluence of two wide parallel

streams of the same density; Exp. 177 91 Depth-averaged concentration distributions dow

n-stream from the confluence of two wide parallel

streams of different density; Exp. 181 92 Depth-averaged concentration distributions, on

arithmetic probability paper, downstream from the

confluence of two wide parallel streams; Flow Sl 94 Cumulative distributions of depth-averaged

concentrations from Exp. 125, f).p/ p = 0 102

a

a) Variance- and b) Coefficient of vari ation-distance curves from experiments with flow SI

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Number 4. 23 4.24 4. 25 4.26 4.27

4. 28

4.29 4.30 4. 31 4.32 4.33 4.34

-xiv-LIST OF FIGURES (Cont'd) Description

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow S 1 and the 12. 5-cm-wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow S 1

and the 20-cm-wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow S 1 and the 10-cm-wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow S 1 and the wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow S2 and the 1-cm-wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow R 1 and the 1-cm-wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow R 1 and the wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow R2 and the 1-cm-wide source

a) Variance- and b) Coefficient of variation-distance curves from experiments with flow R2 and the 20-cm-wide source

Photographs of experiments with flow S 1 and the I-cm-wide source

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-xv-LIST OF FIGURES (Cont'd)

Number Description Page

4. 35 Photographs of experiments with flow R l and the

1-cm-wide source 121

4. 36 Photographs of experiments with flow S l and the

20-cm-wide source 123

·l. 3 7 Photographs of experiments with flow S2 and the

l-cn1-wide source 124

4. 38 Photographs of experiments with flow R 1 and the

wide source 125

4. 39 Photographs of experiments with flow S 1 and the

wide source 126

4. 40 Vertical concentration distributions downstream from a 1-cm-wide source which discharged a fluid with a density the same as the ambient fluid and with a relative concentration of 1. O; flow S2;

Exp. 116 130

4. 41 Vertical concentration distributions downstream from a I-cm-wide source which discharged a

fluid with a density 0.0160 gm/cm3 greater than the ambient fluid and with a relative concentration

of 1. O; flow S2; Exp. 113 131

4. 42 Vertical concentration distributions downstream from a I-cm-wide source which discharged a fluid with a density 0. 0158 gm/cm3 less than the

ambient fluid and with a relative concentration

of 1. O; flow S2; Exp. 128 132

4. 43 Vertical concentration distributions downstream from the confluence of two wide parallel streams

of the same density; flow S 1; Exp. 177 133 4. 44 Vertical concentration distributions downstream

from the confluence of two wide parallel streams

of different density; flow SI; Exp. 181 134 4. 45 Cumulative distributions of particles; a) Exp. 102,

b)Exp.101 136

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Nun1ber

4.48

-:!. 49

4.50

5. 1

5. 2

5. 3

5.4

-xvi-LIST OF FIGURES (Cont'd) De::>cription

Variance -dis ta nee curves for experirn<'nts with

floating particles; flow SI

Variance-distance curves for experinit~nts w.ith

floating particles; (a) flow S2, and (b) flow S3

Variance-distance curves for experiments with

floating particles; (a) flow R 1, and (b) flow R2 Definition sketch for r and b.V

The dimensionless excess variance, D,V, as a function of the dimensionless source strength, Mb, and the dimensionless source width, B

The ditnensionless excess variance, D.V, as a function of the dimensionless source strength,

Md, and the dimensionless source width, B

The intercept, Mt, as a function of the diinensionless source width, B

5. S The fraction of the excess variance, r, as a

function of the dimensionless distance down

-stream, X

5.

6

a) Variance-, and b) Maximun1

concentration-distance curves from some experiments with simi).ar flow conditions and values of Mb but different source widths

5. 7 Laterally averaged coefficient of variation, C , as a function of the dimensionless distance v

6.

1

6. 2

6.3

6.4

downstream, X

Theoretical velocity profiles, w(11) with

w

:.:

0, for three different sets of boundary conditions

Comparison of /:,.V given by Eq. 6. 42 and by experiment

Schematic diagram of the flume used in the MIT experiments (Ref. 2)

Solutions to Eq. 6. 61, -C - dC

+

J fdC) 3

-

·

as

\d

s

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-xvii-LIST OF FIGURES (Cont'd)

Number Description

6. 5 Depth-averaged relative concentration C as a function of

s

,

the dimensionless longitudinal coordinate; for MIT Runs 21 to 24

6. 6 The dimensionless distance s~~ as a function of the parameter J

200

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-xviii-LIST OF TABLES

Nun1ber Description

2.. I Sum.mary of published data on longitudinal diffusion of floating particles in open-channels 2.2

2.3

3. 1

3.2

3. 3

4. 1 4.2

4.3

4.4

5. 1

6. 1

Al

Summary of published data on lateral diffusion of floating particles in open-channels

Summary of published data on lateral diffusion of solutes in open-channels

Specific resistance at 18°C as a function of NaCl concentration

Data from tests to check accuracy of technique for determining relative concentrations

Probability that the error in

cr

2

is exceeded as a function of the number of pafticles collected Sumr:nary of hydraulic data

Li st of experiments with tracer fluids Sample table from Ref 1

Variances of lateral distributions of depth-averaged concentrations in Exp. 125

Lateral turbulent diffusion coefficients from this study

Coefficients in Eq. 6. 30 and 6. 31 for w(T]) and D z Summary of data from experiments with the 1-cm-,

12. 5-cm -, and 20. 0-crn-wide sources

AZ Summ.ary of data from experiments with the wide source

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l

-CHAPTER 1

INTRODUCTION

1. 1 PURPOSE

Many waste effluents which are discharged into streams have densities slightly different from those of the receiving waters. This study investigates the effects of density differences on the horizontal, cross stream mixing of such effluents. Longitudinal mixing under certain restrictive conditions is also examined.

Two typical cases in which the effluents and the receiving waters have different densities are: heated cooling water discharges from industries or steam power plants, and domestic or industrial waste

discharges into brackish estuary waters. In the case of the cooling water discharges, a temperature difference of 10°C causes a density difference of approximately 0. 25 percent. In the second case, a waste discharging into an estuary, the density differences due to differences is concentrations of dissolved salts can be as high as 2. 5 percent. In both these cases the densities of the effluents are less than that of the receiving waters.

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-2-of dis solved matter that are discharged into fresh water streams,

and alsn waste discharges with suspensions of fine particulate matter

t Ii at. i:.; 1nore dense than water. Density differences dne to

con-centration::; of dissolved or suspcn<led 1nattl~r can range up to a few

percent, although'. a fraction of a percent may be h1ore typical.

Alt.hough the magnitude of the density differences gi vcn above

n.1ay secni sn1all, differcncc~s of these magnitudes and snialler often

have large effects on the dynamics of oceans, lakes and the

atmosphere. This study investigates their importance in streams.

1. 2 METHOD OF INVESTIGATION

In this study the effects of density differences on cross stream

n1ixing were studied experimentally in a laboratory flume. In the

c;-.;pednients, which are described in Chapter 3, tracer fluids with

densities equal to and slightly different from the density of the

;:rn1bicnt water in the flunH'! were introduced over some part of the

width and uniformly over the whole depth. If the tracer and

arnbient fluids have the same density, ;crosswise mixing is primarily

by turbulent diffusion. However, if the densities of the tracer and

ambient fluids are different, crosswise mixing is enhanced by a

density-induced secondary flow. The forces driving the secondary

flow arc due to an inbalance of hydrostatic pressure caused by lateral

cknsity gradients. The phenomenon of density-induced circulation is

d Lscussed in more detail in the literature review of Chapter 2 and in

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.

,

-· '

-So111t· of t.111~ clcd .. l, sliuwing l:ht: charact:ori.:~t:ic r~.·-d.111·c11 of the c,>11~cntt·d.tion di::;triliution:-; obsc1·ved in the t•xperin1e11ts, arc given in Chapter 4. The concentrations n1easured at all points in all

experin1ents are made available in Ref. 1, but they are not included in this text because the data are too numerous.

A dimensional analysis of the problem of lateral mixing in

an open channel is given in Chapter 5. The results of the analysis and the experimental data yield empirical curves for parameters which

characterize the tracer-plume width and the variation of concentration

wi l:h depth as fnnctions of distance downstrcan1 frorn tl1e source, the

ini lial density difference, the source width, and the hydraulic

para1neters of the stream. Concentration distributions can be

estimated by using these curves.

A simplified theory for the effect of horizontal density gradients on horizontal mixing is given in Chapter.

6.

The velocity distribution

c:.:i..used by a horizontal density gradient was derived, and an

e:~pression for the dispersion coefficient due to the density-induced velocity was calculated. An analysis using this dispersion coefficient

p1·edicb: that the characteristic width of a tracer plurn1· in a.n open

were obtained from the diinensional analysis and data. However,

because some of the simplifying assumptions rnade in the derivation of

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-4-The approximations in the derivation of the density-induced velocities are more nearly correct for some experiments in an idealized

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-5

-·CHAPTER 2

BACKGROUND AND REVIEW OF PREVIOUS WORK

)

,

_

.

Tl! E CONSEHV ATION EQUATION

.In most n1athematical analysis of turbulent flows, the transport

of solutes or other tracers is often described by an appropriate

time-averaged conservation equation for the tracer. The equation usually

used for aqueous solutions, which are nearly incompressible but not

necessarily of uniform density, is

(2. 1)

(see e. g. Harleman (3)). In this expression, tis time; c is a time

-avcra.µyd concentration: u, v, and w arc tin1c-averagcd velocitie8 in

the rc•cl:ilinear coordinate directiorrn, x, y, and :1.; a11d e , r-: , and ,.;

x y z

a:re turbulent diffusion coefficients for mass. (The velocities and

concentrations in Eq. 2. 1 are averages over a time that is long

compared to the time scale for turbulence, but short compared to

the time scale for the gross phenomena being investigated. )

If the density of the fluid is a function of c, then u, v, w,

E: and r:: may be functions of c also. Under these conditions,

'y' z

€ ,

x

Eq. 2. 1 is nonlinear in c and must be solved simultaneously with

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-6-does not affect the dynamics of the flow, Eq. 2. I is linear in c and the equations of n1otion for the fluid n1ay be solved separately. However, even under these simplified conditions, analytic solutions for Eq. 2. 1 have been obtained only for very simple flow fields.

Below, a review is given of existing information on mass transport in turbulent open-channel flows of homogeneous density. It is followed by a discussion of the effects of density differences on mass transport.

2. 2 MASS TRANSPORT IN OPEN-CHANNEL FLOWS OF HOMOGENEOUSDENfilTY

2. 2. 1 Longitudinal Dispersion.- The coordinate system for an open channel flow is chosen with the origin on the channel bottom, the x-axis in the direction of flow, the y-axis normal to the bottom

and positive upwards, and the z-axis horizontal and in the lateral direction. For a uniform steady flow of a fluid of uniform density in a wide channel of uniform depth, v

=

w

= o,

and ae:

I

ax

=

ae

I az

=

o.

x z

For these conditions, Eq. 2. 1 becomes

(2. 2)

One now substitutes into Eq. 2. 2 the expressions: u =

u

+

u'

c

=

c

+

c'

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-7-where the overbarred quantities are depth-averaged variables and the primed quantities denote deviations from the averages. Averaging the resulting expression over the depth, and recognizing that

au I

I

ox

::::

O, yields

(2. 3)

The second term on the left-hand side of Eq. 2. 3 represents

differential convection by the mean velocity,

u,

and the third term represents differential convection due to the correlation between u'

and c'. Elder (4) showed that for two-dimensional flows in which

fk/oz

=

O, one can write for large time

u'c'

=

-D Ile

x

ax '

(2. 4)

where D is called the longitudinal dispersion coefficient.

x He showed

that D is only a function of u' and € and is given by

x y

1 T\ T\1

Dx

=

-d2

J

u' (T))

J

€ tri•)

J

u' (T\11) dT) dri'

d

n"

,

0 0 y 0

(2. 5)

where T), ri', and T)11 are y Id. Using the Prandtl-von Karman

logarithmic velocity distribution to obtain u' and the Reynolds analogy to obtain € (see Subsection 2. 2. 3 ), Elder integrated Eq. 2. 5 to

y give

where k is von Karman's constant; u* is the shear velocity, and d

(2. 6)

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-8-2. -8-2. 2 Longitudinal Diffusion. Because transport by longitudinal

dispersion, as described above, and transport in the longitudinal

direction by turbulent diffusion are additive, and because D

>

E:

x x'

little experimental data on e exist.

x

Data on longitudinal diffusion at the free surface are available

from experiments by Sayre and Chang (5) and by Engelund (6 ). In

these studies, small floating particles were released at a point on

the free surface, and either the distribution in time for particles to

travel a fixed distance or the longitudinal distribution of particles

after a fixed time from release were observed. Longitudinal

turbulent diffusion coefficients at the free surface,

e

,

were XS

calculated using

E:

XS (2. 7 a)

or

(2. 7b)

where cr~ and CJ~ are the variances of the distributions of the particles

m space and time, and u is the mean longitudinal particle velocity.

s

Sayre and Chang normalized E:xs by dividing by u,;,d, but

Engelund chose to divide his coefficients by u d. Both normalizations s

were made on both sets of data and the results are given in Table 2. I.

Neither normalization of the published data give·s more consistent results

than the other; however, using the re-evaluated coefficient for Engelund1_s

Run B, which is obtained by recomputing E: from his published basic xz

data, the normalization by u,~d gives slightly more consistent results.

Normalization by U:*d is also more appropriate because u,." characterizes

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Table 2. 1 Summary of published data on longitudinal diffusion of floating particles in open channels.

Reference Run No. Channel

Engelund (6) A Laboratory flume

230 cm wide; 2. 3 cm diam. eand

roughneu on bottom

B Laboratory flume

230 cm wide; O. l cm diam. •and roughneu on bottom

Sayre and LO-P-1 Laboratory flume

Chang (5) 239 Cm wide;

rough-LO-P-2 neu blodte 1. 9 cm

LO-P-31

high. 7. 6 cm wide, l. 6 cm long on bottom

1 _{Average-;;! two eJl:l>erimente.} a Published data.

3 _{Re-evaluated data.}

Floating Depth _VelocityShear Particles d

u* (cm)

(cm/sec)

Plastic ball 5.45 3. 60

0. 9 cm diam sp. gr. = O. 288

Plaetic ball 17. 3 l. 60

O. 9 cm diam. •P· gr. = O. 288

Polyethylene diace 14. 8 3.81 O. 3Z cm diam.

0. 16 cm thick Z4.8 4.90

sp. gr. = O. 96

37. 1 6.04

Ave. Water Particle Diffusion

I

Velocity Velocity Coefficient £XS £ _XS

\l u _s CX& u*d Ud ₈

(cm/eec) (cm/sec) (cm2 _/_sec)

30.0 38. 3 10.0 0.510 0.0479

30.6 34. 4 25. 0 3 0. 903 ~ o. 0420 3 (18. 5 )3

(0. 668 )3

(0. 031 03

23. 5 33.S 30.7 o. 5-l4 0.0639 34. 4 45.5 57.0 0.470 0.0505

47.5 65. 9 23.8 l. 06 o. 0975

I

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-1 ()

-Cotnpari11.t~ th(~ data in Table l. 1 wit;1 Eq. 2. 6 <·crnfir111~: I.hat.

].) : .... > 1·: x x

2. 2. 3 .Lateral Diffusion

2. 2. 3. 1 Diffusion at the Free Surface. Data from experiments with floating particles have also been used to obtain values of E: , the lateral diffusion coefficient at the free surface.

ZS

Experiments of this type were first made by Or lob (7 ), later by other

investigators, and also in this study. A summary of the experimental

data from. these investigations is given in Table 2. 2. Values of E:zs

were calculated by either the formula

or

da:a

1 s E:zs

=

Zat

E: =

ZS

u da2

·S S

2 dx

(2. 8a)

(2.8b)

where (J2 is the variance of the lateral distribution of particles at a

s

fixed time after release or after traveling a fixed distance. The

dim.ensionless coefficient a is defined by s

(2. 9)

The average values of a. from each of the studies listed in Table 2. 2

s

range from O. 172 to 0. 241. They are less than the normalized

longitudinal coefficients given in Table 2. 1.

2. 2. 3. 2 Depth-Averaged Diffusion Coefficient. Because

(29)

Table 2. 2 Summary of published data on lateral diffusion of floating particles in open channels.

Depth Shear Water Particle Reference Run No. Channel _ParticlesFloating _d Velocity Velocity Velocity

u* ii' u

(cm) I

(cm/sec) (cm/aec) (cm/sec)

Orlob (7) Laboratory flume Polyethylene disc• l. 62 122 cm wide, 0. 32 cm diam.

to expanded metal screen 0. 16 cm thick

12.8 roughness on bottom ep. gr. =0. 975

Sayre and Laboratory flume Polyethylene discs

Chamberlain (8) 239 cm wide, O. 32 cm diam. 17.4 3. 51 49.8 62.5 sa_nd dunes on bottom O. 16 cm thick

ep.gr.=0.96

Sayre and LA-P-1 Laboratory flume Polyethylene discs 14. 8 3. 81 23.5 33.5 Chang (5) 239 cm wide; roughne11 0. 32 cm diam.

LA-P-2 blocks 1. 9 cm high,

·

o

:

16 cm thick 24.8 4.90 34.4 45. 5 LA-P-3 7. 6 cm wide, •P· gr. =O. 96 37. I 6.04 47.5 65. 9

I. 6 cm long on bottom

Engelund (6) A Laboratory flume Plutic ball 5.45 3.60 30.0 38.3 230 cm wide; 0.9 cm diam.

2. 3 cm diam. sand ep.gr.=0.288 roughness on bottom

B Laboratory flume Plaetic ball 17.3 l. 60 30.6 34.4 230 cm wide; O. 9 cm diam.

O. l cm diam. sand ep.gr.=0.288 roughne u on bottom

Thie study Laboratory flume Polyethylene 3.90 110 cm wide; particlee about

smooth bottom or O. 35 cm diam. to expanded metal screen sp. gr. =O. 95 11. I roughness on bottom

1

Averag., o! lZO experiments. The values of u* and d were corrected for blockage as de1cribed in Subsection 4. l. 3. 3 _{Average o!}₁₃_experiments. _Data_for_all_{experimenta are given}_{in Tables}_4.₁_and_5._l_.

Diffusion

Coefficient 'zs

E: _ZI

a,=

u*d

(cm3 _/sec)

o. 172 1

15. 0 0.241

13. 3 0.236 23.4 _o.196 59.2 0.264

--

_0.₂₃₂

4.0 0.204

6.5 0.234

-0.219

(30)

-12-t. _z, the depth-averaged value of the lateral turbulent diffusion _. _. coefficient. This coefficient is usually calculated with the formula

or

E:

z

where o2 _is_the_{variance of the lateral distribution of the} depth-averaged concentration,

c,

and o~ is the variance of the lateral distribution of

ac

I

az.

(2. lOa)

(2. lOb)

Eq. 2. lOa and b are obtained from Eq. 2. 3 as follows. Far

downstream from a source of a neutrally buoyant tracer, experimental data show that c' <<·c; therefore, one can delete each of the terms in Eq. 2. 3 that contain c' because there are corresponding similar terms containing

c.

One can also delete the

term~

aac

because it is less than

x x

a

-the deleted term

ax

(u'c' ). Therefore, for a steady source Eq. 2. 3

becon1es

-

a-c

- a

2

c-u -

ax

=

e: _z8z2- - · (2. 11)

Taking moments of this expression in the z-direction yields Eq. 2. lOa, or taking the derivative of Eq. 2. 11 with respect to z and then taking

moments yields Eq. 2. lOb.

Values of

e

and the dimensionless coefficient z

e

_z Cl = Uo!<d

from a number of investigations are given in Table 2. 3.

(2. 12)

These data can

(31)

-13-Table 2. 3 Sum1nary of published data on lateral diffusion of solutes in open channels.·

Elder (4)

Sayre a.nd

Ch&J?I (!>)

Sullivan (9)

Olover a,.•

110. p. 26)

Fi•cher ( 11)

Yatau.kura~

Flacher,

a.nd Sayre ( 12)

Fiecher (13)

Okoye (14)

Thi• etudy

Run No.

Laboratory flume

lS. 5 cm wide; amooth bottom.

Trac•r Pota•elum permanaan•k dye Concentration• D•t.rmlned by Depth d

Analyeh of ... 5 photo1raph• with

mlcrodanettomater

LA-D-1 Laboratory Ou.ma .Fluoreecent ContinUOlH

eamplin1

fiuororneter

14. 8 239 cm wide; rouahn••• dye•

L.A-D-2 block• 1. 9 cm hiah, 24. s

37. I 1. 6 cm wide

LA-D-3 I. 6 cm lona on bottom.

6/21 6/23

Laboratory fiwna,

76 cm wide; amooth bottom.

Analyeta of

pboto1raphe with

microd.enattometer

10.2 8. 9S 7. ]]

Laboratory Own•• Salt Meuureanent of fluid conducd vity

~~

14. 7

13. 8 28. 8 Z<&Z cm and IZZ cm

wide, rO\l&b and emootb

bottom•.

Colwnbla River near

Richland. Waeh.; approx 300 m wide

Radlonucllde• Radiation cou.ntin.1

ta a coolln1 of •ample•

water dlecbu1e

Atrl•co Fe9der Canal Rbodamlne WT Analy•t• of aample• 68. J

with tluorometer

near Bernalillo. N. M.; dye approx 17. S m widei

30-cm-hlah, •and dune•

on bottom atriU1ht reach

66. 7

Mbaourt River Rbodamlne B Analyeh of •amplti• ...270

with tluorometer

nea.r Blair, Nabruka dye a.pproa. ZZS m wide.

Curved laboratory Ahodaml.ne WT Analyete of •ample• 3. OZ channel: 76. l cm dye with tluorometer 5. 28

l. 72 2. 03 2. 20

wide, •mooth

bot:tnm.

La.bora.tory flwne•

110 cm 'fl'ida and

85 cm wt.de; unooth and 1ravel bottoma.

Laboratory tlume

110 cm wide;

•mooth and rou1h

bottom•.

Sodium chloride Meaaurecnent of

fluid conductivity

!!:!~

5odlwn chloride Mea•urement ol

apecillc re•latance

of aamplea

1.6

lo

21. I

l. 90

lo

11.l

Shear Water Oiffu•lon Velocity V•loclty Coeftlcient

u. u c.

(czn/••ol (cm/•ec) (cm• /••c)

3. 81 4. 90

6. 04

0. 827

o. 988

1.21

6. 29

6. 13

7.4

2. 66

1. 70

2. 13

I. )7

I. 68 23. s 34 ... 31. l

ts. 1

18. s

22. 9

134

63. s

66. I

... 11s

31. 7 27. 0

26. 8

19. 0 19. 7

9. SB

zo. 2 36. 9 0. 902 O. 97a

l. 182

H.S 8. 28 7. 16

1860

102 102

1200

11. 3

z I. ..

10.6

l. 96 I. 88

1

Eld•r1

• 1"-•hlhhiect Averaee vah.ae, o. ~ O. 228, waa multiplied by 0/1, ll)• becau•e A..:corJtn1 to 9.iWvan (9), Elder uaad half the diatance betw•en point• where &he concentration• ware h...U the pealt Yalue tn place of O tn Eq. 2. 10a.

• Su.lUyan'• report•d v&J1J•• hav• b••n divided by Z bec•u•• h• omitted the factor t ln i:.q. z. lOa.

~ Tr•cer• were not nautrall.f buoyant.

• Hydraulic radlue u•ed tn pl•c• of d•pth for compu.tin1 Q.

6 _Averaae_{of 13 e.xperbnea.t•}_. _Data_for_all_{ex:perim.ente are}_{atvan in Table•}_'_·₁_{and S}_._1.

o. 164 l

0. 170

o. 17'1

o. 160

0. 167

O. 107 a

0. 110 a

0. 133 2

0. 117 0. 36

o. 22

0. 14 o. 12

a. z4

0. ZS

0. 6

I. 4 2. 4

I. l

o. 70 0.1151

.. o. 14

(32)

-14-(Ref. 4, 5, 9, 10, 11, 14, this study) and the other from experiments in curved channels (Ref. 10, 12, 13). Except for the data by Glover, which were fron1 experin1ents in which the tracer was not neutrally buoyant, the data from straight channels yield values of

a

between O. 10 and 0. 25. (One should observe that the often quoted value published by Elder (4) has been corrected in Table 2. 3). All v a 1 u e s from curved ch an n e 1 s are greater than 0. 5. The higher values in curved channels have been explained by Fischer ( 13)

as due to secondary currents induced by the bends. These currents transport material in the lateral direction by a process analogous to longitudinal dispersion as described in Subsection 2. 2. 1. Fischer was able to calculate a lateral dispersion coefficient using Eq. 2. 5 with lateral velocity deviations, w', in place of the longitudinal

velocity deviations, u 1 •

A suitable explanation for the variation in

a

for straight channels does not yet exist. However, it is worth noting that the range in

a

found by one investigator is usually not as large as the range of the means of values given by different investigators. Therefore, much of the variation may be due to weak secondary currents characteristic

of the different flumes or channels. Attempts to correlate

a

to the width-to-depth ratio of the flow cross-section and to other parameters is presently being investigated by Okoye (14).

In both the study by Sayre and Chang (5) and in this study, where

experiments were made to obtain both a and a , one finds that a is

s s

(33)

-15-the difference between a and as was sn1all if they used

u

in place of u in Eq. 2. 8, the data from this study do not confirm this re~;ult.

s

2. 2. 4 Vertical Diffusion. An estimate of the vertical diffusion coefficient for mass in a wide open-channel flow, € , can be obtained

y

from the Reynolds analogy. One first expresses 'fxy' the apparent shear stress on a horizontal plane1 as

'f - (\ € )

au

xy - y y p

oy ,

(2. 13)

where p is the fluid density, and the product A. € is the eddy viscosity. y y

The term

t.

is the turbulent Schmidt number and is the ratio of y

the diffusion coefficients for momentum to mass. Using the Prandtl-von Karman logarithmic velocity distribution,

u

,:c

u(y) - u(d)

=

k

tn~, (2. 14)

and a linear distribution of shear stress,

(2. 15)

Eq. 2. 13 yields

(2. 16)

This expression for A. t. is zero at y

=

0 and y = d, and is symmetric y y

d

about y

=z

·

The depth-averaged value is given by k

\yey

=

6

u*d . (2. 17)

Vanoni ( 15) observed the vertical distributions of suspended sedi-ment in an open-channel flow and.used the data to calculate e for ·sediment.

y

(34)

-16-and that A ~ 1. Jobson and Sayre (16) introduced dye at the surface

y

of an open-channel flow and observed the vertical distributions of

dye as a function of distance downstream. They used the data to

calculate i:: and also found that E: and A E: were sirnilarly distributed

y y y y

and that >.. :-"'3 1. Similar experiments with fine sediment gave nearly

y

1 '1<' t;;11111· result!~.

/... 1 1~:Ft•'l•~CTS OF DENSITY DIFFEHENCES ON MASS THANSFEH

2. 3. 1 Vertical Density Gradients. Observations of vertical

tur-bulent diffusion in the atmosphere, in natural water bodies, and in the

laboratory show that stable vertical density gradients reduce vertical

diffusion (Ref. 17 and 18). Observations also show that the

vertical diffusion of mass or heat is reduced more than the diffusion

of momentum. Taylor ( 19) noticed that stable density gradients in

the atnwsphere reduce turbulent wind velocity fluctuations in all three

dir.ect:iom;; hence, one n1ay suspect that a stable vertical density

).;.radil.'nt. J1"1ay r1.~d11c<· turb1d.cnt diffui,;ion in the hori.zontal direction

also.

Ellison and Turner ( 18) have given a physical explanation for the

reduction in vertical diffusion by a stable density gradient. They

reasoned that a parcel of fluid which is displaced from its equilibrium

postition by turbulence m ay return to its equilibrium position before

it mixes completely with the surrounding fluid. They also reason that

the reduction in the diffusion of momentum is not as great as the

(35)

-17-exchange m01nentum without .mixing by the action of pressure, but jn order to transfer heat or mass the parcel of fluid must mix with its

surroundings.

Most investigators attempt to relate the decrease in diffusion caused by a stable density gradient to Ri, the Richardson number,

Ri

=

g_

~

/ (

au)a

p

Hy

Cly ' (2. 18)

whcrv !-'. is the ~cceleration due to gravity, and 8u/()y lA the vertical gradient of the primary flow which is considered to be horizontal. Munk and Anderson (17) suggest the empirical equations:

e A.

=

(e A. )₀ (l +Ri)-t

y y y y (2. l 9)

and

e

=

(e )o (1+130

Ri)-~,

y

(2. 20)

where (e: _y)., _y)₀ and (t. _y)0 are the values of

e

_{y y}A. and

e

_yunder neutral conditions, Ri -= 0. Okubo (20) reviews formulae suggested by other

i.1 tVf' H tig;ttnr H.

Because the longitudinal dispersion coefficient, D , given by x

Eq. 2. 5 is inversely proportional to

e

,

one may expect that D would

y

.

x

be larger for stable density-stratified flows than for flows of uniform density.

2. 3. 2 Horizontal Density Gradients. Observations show that density differences in the horizontal direction can increase mass

(36)

-18-homogeneous turbulent field. They found that the rate of mixing of the two fluids increased with increasing density difference. Jen, Wiegel,

and Mobarek (22) and Hayashi and Shuto (23) performed experiments with heated jets of water that discharged horizontally near a free

surface; they found that lateral spreading of the jets increased with increasing difference in temperature between the discharged and ambient fluids.

Density-induced velocities also occur in estuaries, where long i-tudinal density gradients exist due to the difference in density between salt water frorn the ocean and fresh water from rivers. Experiments in an idealized laboratory estuary by Ippen, Harleman, and Lin (2 ),

which are analysed and are described in more detail in Chapter 6, show that as the difference in density between the water in the ocean and

incoming river increases, the apparent longitudinal diffusion coefficient increases also. Ippen (24) has explained conceptually the effect of

longitudinal density gradients on the velocity distributions in estuaries, and Hansen and Rattray (25) have treated the problem analytically. If

the pressure distribution over the vertical is hydrostatic, and if the

estuary is well nlixcd vertically so that the variation in density with depth is small, the pressure is given by

p

=

(d-y) pg.

If the x-axis is horizontal, the longitudinal pressure gradient is then

given by

(y-d) g *

+

Spg =

- .££

_ax _, (2. 21)

(37)

-19-vertical distribution of each of the terms in Eq. 2. 21 are shown in Fig. 2. 1 for a seaward sloping water surface. Because the pressure gradient due to the water surface slope is invariant with depth, and because the pressure gradient due to the density gradient is zero at the water surface and reaches a maximum at the botto1n, the resultant pressure gradient near the surface and near the bottom can be in opposite directions. The result can be an upstream flow near the

bottom but a downstream flow near the surface. When used in Eq. 2. 5, velocity distributions of this type yield higher longitudinal diffusion coefficients than unidirectional velocity distributions with the same xnean velocity. Because the upstream flow in the lower layer must

return seaward in the upper layer, the flow pattern resembles a longitudinal circulation of the primary flow.

The present study investigates the effects of density-induced secondary circulation on lateral mixing in turbulent open-channel flows with lateral density gradients.

y

ap

(y-d) g

ax

+

Spg

=

"""

\

'

. ./

(38)

-20-CHAPTER 3

EXPERIMENTAL APPARATUS AND TECHNIQUES

3. 1 DESCRIPTION OF EXPERIMENTS

Experiments were n1ade in a laboratory flume to observe the

lateral mixing of fluids with densities different from that of the

ambient flow. The objective of the experiments was to obtain data

for developing and checking techniques for predicting horizontal

mixing rates in turbulent flows with horizontal density gradients.

In the experiments, a tracer fluid was introduced continuously

into the flume uniformly over the depth and across some fraction

of the flume width. The tracer fluid was introduced into the flume at

the same velocity as the ambient flow in the flume. In most

experiments, the source was located on the flume center line, and the width of the source was small compared with the width of the flume.

However, in some experiments that were made to observe the

mixing of two wide parallel streams, the tracer fluid inihally occupied all that part of the cross -section on one side of the flume center line.

Most experiments were made with tracer fluids whose densities

were slightly greater or the same as the density of the ambient water

(39)

-21-fluids which were less dense than th() ambient water. All tracer

fluids were salt water (NaCl) solutions. The densities of the heavy

tracer fluids were varied by changing the salt concentrations. Tracer

fluids of neutral density were prepared by adding methyl alcohol to

a salt-water solution. The light tracer fluids were made by adding

additional amounts of alcohol. The tracer fluids were often colored

with an organic dye to make them visible.

Distributions of tracer fluid in cross -sections downstream from

a source were obtained by taking water samples from points in the

cross- sections and by measuring the specific electrical resista~ce

of each sample to determine the tracer concentration in each sample.

The lateral diffusion of small floats was observed during

experi-ments in which the tracer fluid was neutrally buoyant and also in

experiments in which no tracer fluid was introduced into the flume.

The purpose of these experiments was to provide additional data to

compare the turbulent diffusing characteristics of the flows in the

flumes used in this study with the flows in the flumes used by other

investigators.

3. 2 FLUME

3. 2. 1 Description. The experiments were conducted in a

40-meter-long recirculating flume located in the sub-basement of the

W. M. Keck Laboratory of Hydraulics and Water Resources.

Fabrication details of the flume are given by Vanoni, Brooks, and

(40)

.,

-

..

.,

..

,

"

.

..

'·~'--- . I

'·

0 ; ! I 110.0 cm (INSIDE)

[1

- r--- -- ---1

I

I -tl- ~f- '

11 I; l•r = =9 I . •: .

. I I !•

; Ci ~' r----_J " ! ;

I

_I !! ~ _{J ' . !}_____;

.-

; )

}

' :· ·

;

,

.

i 1

L ___ - - --- _j ._ _ _ .J ---,~-'

WEIR__/ . ~ .:

!._ - - --- - - -- - - --- ---- 38.57m - -- -- - -- - - -- -, ~---R-E=OtR ::::::.7---_-_-:_-_-_-:_-:_-_-;.-;_"0--:-_-:_-:_~·-;

- INLET TANK

STAINLESS STEEL

-PLATE

f SCREEN SECTION

1- STAINLESS STEEL

i SIDES

-,:'W </

\

i RllBSER. PIPE \_JACK

SECTION A-A ENLARGED X3

A

VENTURI METERS .

i i - 3.007m- , ---· -·-49 a534m--- -·---1

GLASS SIDES ~.

\

/. . .t". / j ,;~/-~/

L JACIC DRIVE SHAFT

- - -- - - --- - - -- ---- - - -- -· 4084m- - -- - - I

STAINLESS STEEL SIDES-,

\

' I

RESERVOIRS"?>.,.

! , ..

OUTLET-,

TAM<

!

I

r -3£96m-

-·

..

·

...

NOTE: ARROWS---DENOTE FLOW PATH

Figure 3. 1 The 40-meter flume. (Drawing modified from Vanoni et al. (26) ).

I N

(41)

-23-in the photograph in Fig. 3. 7. The bottom of the flume is made of stainless steel plate and is 110 cm wide; the vertical sidewalls arc

(il cn1 high and are glass alollg most of the flunw's length. An instrument carriage rides on rails which are n10unted on the tops of the sidewalls.

The flume is supported in pivot bearings near its mid-point and is supported elsewhere by four pairs of power-driven screw jacks

(as shown in Fig. 3. 1 ). The flume slope is obtained by reading a vertical staff attached to the flume approximately 17 meters upstream from the pivot. The staff was calibrated by observing the slope of

the flume bottom with respect to a still water level.

3. 2. 2 Water Circulation. Water entered the upstream end of the flume through a metal screen (8 mesh/ inch) from a baffled inlet tank. An adjustable weir was installed at the downstream end of the flume to control the tail-water depth. Water flowed over the weir into an outlet tank.

The design of the flume permits a number of alternatives for returning the water fron1 the outlet tank to the inlet tank. In order to n1inimize changes in the tracer concentration of the ambient

flow during an experiment, the return system was modified to utilize the maximum water volume. Water flowed from the outlet

tank through a pipe and a laboratory floor drain into the farthest of the four reservoirs. The reservoirs are connected in series by short lengths of 10-inch pipes. The reservoirs have a combined

(42)

-24-frorn the last reservoir, which is under the outlet tank, through a nominal 8-inch pipe, which runs under the flume, to the inlet tank.

The pump was driven by a IO-horsepower motor through a contin-uously variable V-belt drive. Water discharge was measured with

a venturi meter (Keck Lab. No. Q-6) in the 8-inch pipe. The 16-inch pipe which runs beneath the flume was not used for the return flow and was capped near the outlet tank but was allowed to fill with water from the inlet tank.

3. 2. 3 Rough Bottom. The bottom of the flume was made rough for s01ne experiments by laying expanded metal sheets (Fig. 3. 2) on

the stainless steel bottom. The sheets, which are manufactured for use as plastering lath, are made of steel and are galvanized; their approximate dimensions are given in Fig. 3. 3. Although the total thickness of the lath is 0.31 cm,the solid volume is equivalent to that of a solid sheet only 0. 025 cm thick.

The lath was cut into panels 73 cm by 109 cm which were installed with the 109 cm length spanning the width of the flume.

The panels were held to the stainless steel botom with mastic stripping 1 /2 inches wide by 1 /8 inches thick. Single strips of mastic were laid about 5 cm from each wall along the entire length of the flun1.e, and three pieces about 8 cm long were laid longitudin-ally at the joints between panels. The panels were pressed into the mastic so that they lay flat on the stainless steel bottom.

(43)

-25-· flow direction

scale reads in centimeters IJll/ !l,1111/llll/Jllljlllllllllllllllllll\llll\llll\111111111\ll\1\llll\1111\ I

6 7 8 9 10 l l 12 13

1 0 1 1 6

Figure 3. 2 Photograph of expanded metal lath used for roughness on the bottom of the flume.

O.Blcm

E

g

~ ~.,,

"-~r

_{m f}

ouumed bottom stoinleu steel bottom

SECTION A-A

Figure 3. 3 Dimensioned sketch of metal lath used for

(44)

-26-3. 3 WATER DEPTH AND VELOCITY MEASURING EQUIPMENT

Water depths in the fluine were measured with a point gage

which could be read to the ncarcHt 0. 01 ctn and which waH rnountt-~cr

on the instrurnent carriage.

Water velocities were measured with a 1 /8 inch (0. 32-cm)

diameter pitot- static tube. The differences in pressure between

the dynamic and static ports were measured electronically with

a pressure transducer and strip-chart recorder. The system was

calibrated by imposing pressures on the transducer with a water

manometer which could be read to within O. 001 inches (0. 0025 cm).

The difference between calibrations before and after an experiment

was always less than two percent.

The pitot-static tube was attached to the traveler of a point

gage on the instrument carriage for positioning it in the flume. To

obtain the velocity at a point,the output of the pres sure transducer

was recorded for 30 seconds and was averaged by eye.

3. 4 TRACER FLUID SYSTEM

3. 4. 1 Fluid Handling System. The tracer fluids were prepared

in a mixing tank and were pumped into the flume through various

types of sources which are described in Section 3. 4. 2. The mixing

tank was open at the top and had a capacity of approximately 600 liters.

The tank and associated plumbing are shown schematically in Fig. 3. 4.

The mixing tank was filled with water from the 16-inch

(45)

SHUTOFF

VALVE

I I ~ YZ . . . -~---i'O'...----~

TO SOURCE

REGULATING )If

VALVE

I

IN FLUME

VALVE

PUMP

FLOOR

I

01

I

FLOWMETER

t

•

I

VALVES

MIXING

TANK

· - - 4-•yx=i. n

- - - . . _ , ~JS'Sll ,5,;p

·. ··:<:>:

~·

·.

:~.:

·:

·:.~:-.: :·:·!>.::~.:: :.~.":. :·~:~.J

r

:

j;:

A··

.

-:

·

..

:.-::·~

:-

:.::

'

...

~:A"·:.:4;-.;~ ~;

:

o

.

~.·,.A . •

·.'!'

.

•

..

.o .. -.~.·.A.·

.

.•..

.

.. ·

.

· .. ,

~·.:

.

..

~-

.

...

·

.·

..

·

.

. · ,

.

I

..---

--

---i'

1---....,..

16- INCH PIPE

Figure 3. 4 Schematic drawing of the tracer-fluid handling system.

,

(46)

-28-alcohol (if any) were added to the tank and were mixed by pumping water out of the bottom of the tank and returning it at the top.

For soni.e experiments dye was also added to the solution.

During most experiments, fluid was pumped from the rnixing tank to the source in the flume. However, for some experiments, wat·er with no tracer and with the sam e density as the flu.id in the

flt11.111· w:IN p1111q>1:d din •c.,'.Ll.y ft•t1nl tlu· J(, .. inch pip1~ l.o I.hi: 1-101n·cc.

Tht~ .r.low rate tn tlie source was ineasured with a variable area

flown1eter (Keck Lab. No. Q-29) and was regulated with a gate

valve. A shutoff ball valve was put in series with the regulating valve for convenience.

3. 4. 2. Sources. The tracer sources in the flume were designed

to distribute the tracer uniformly over the depth and across some

selected width within the flume. They were also designed to cause rninimun1 disturbances to the ambient flow. Three designs were

11:-;ed: 011c design was used for the narrow sources, cm and 2 cm ,vidc; a second design was used for the 12. 5-cm - and 20-cm-wide i.nternicdiate width sonrces; and the third design was used for the

wide source in which the tracer was distributed over one-half of

the 110-cm-wide flume.

3. 4. 2. 1 Narrow Sources. The narrow I-cm-wide source, which is shown in Fig. 3. 5, was made by placing a vertical manifold between two vertical guide walls. The manifold was a

flath med length of thin-walled brass tubing with holes drilled in

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-29-1 OT09.

Figure 3. 5 The 1-cm-wide source.

10114

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-30-0. 3 cn1. intervals along a vertical line starting near the botto1n and

extending about 11 cm high. Holes above the water surface were cove:r.ed with plastic tapE:~. The holes were 1nadf'. with a l /8-inch

-di;U.))1::t,•1.· drill and overy th'i1·rl hole: w;ui t•nl;1rged t:o a J/1(1-

inch-di.:irnch'r. A µiece of rolled wire screen was placed in front of the manifold to break up the jets from the manifold.

The vertical guide walls were made of 24-gage (0. 028-inch)

galvani:;,ed steel sheet. They were 1 cm apart and parallel at the

downstrea1n end but met to form a sharp edge at the upstream end.

The total length of this source was 25 cm. The surface waves generated by this shape were less than 0. 15 cm high and were not

much larger than the waves 1nade by a single piece of sheet metal

placed parallel to the flow.

The di::;cha.rge fron1 the 1-crn-wicle source was equal to the

dischar~e through a l-cn1-wide vertical strip in the central part

of the flltmc. The velocity distribution at the mouth of the source was oh8erved to be approxim.atcly uniform over the depth. The downstream end of this source was located 15 meters downstream

from the screen in the inlet box of the flume.

A 2-cm-wide source, 61 cm long, was constructed similar to

the I-cm-wide source. However, the 2-cm-wide source was·us.ed

only in an experiment to observe the lateral diffusion of floating

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-31-3. 4. 2. 2 The Intermediate Width Sources. The 12. 5-crn-·

and 20 -crn-wide sources were n1adc by placing a horizontal manifold between two parallel vertical guide walls. The guide walls were open at each end to allow the watc:~r in the flunw to flow l.rntwec n then1. A tracer fluid fron1 the mixing tank was sprayecl by the inanifold onto the surface of the water flowing between the guide walls. This design did not require mixing, storing, and pumping large volumes of water as would be required if the sources were constructed in the same way as the narrow sources. The 20-cm-wide source is shown in Fig. 3. 6.

The sa1ne guide walls were used for both sources. They were 9 J.,5 crn long, 20. 3 cm high, and were made from 18- gage (0. 049-inch) stainless steel sheet. The walls were fastened to metal rods of the proper length which held them apart and upright. The contact between the guide walls and the rough lath bottom was sealed with mastic. Plastic tape was used to seal the contact between the guide walls and the smooth bottom.

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-32-cross-piece. The holes were spaced 1. 25 cn1 apart in lhe nianifol<l

for the 20-cni-wide source, and 0. 65 cn1 apart in the 1nanifold. for

the 12. 5-cin-wide source. The holes were approximately O. 25 crn

in diarnt•tcr; thl~ holes were niadc sli1i;htJy larger nc<.1r !:he ct•ntc:r

a1HJ s1nalh'!r 11<::.n· the C'nd ~; i.n ord<~r t.o 1nake the tracer concn11tratio11

distribution n1ore nearly uniforin across the source.

Flow rates through these manifolds were the same or less than

the flow rates through the manifold in the narrow I-cm-wide

source. The combination of these flow rates and manifold port

sizes produced jets which penetrated the water to give sufficient

rnixing over the depth but did not excessively disturb the flow. The

manifolds were tilted slightly to give the jets a small downstream

con1ponent of rr\01nentun1 to oppose the hydraulic resistance of the

~uide waUs on the flow between them. The downstrean1. ends of the .

gltide walls for both these sources were ] 7. 8 n1eters downstrean1

fron1 the screen in the inlet box.

Almost all experiments with the 1-cm-, 12. 5-cm-, and 20-c

m-wide sources were conducted with the sources on the center line of

the flume. To test the applicability of the data from the experiments

to proble1ns in which an effluent is discharged along the bank of a

stream, a 10-crn-wide source was placed against the left sidewall

of the flun1e. Data fro1n experiments with this source were analysed

as if the wall were a line of symmetry for the 20-cm-wide source.

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-33

-111 lhl):.;c 11~:c:d i.:1 lhc ll.. r;_ ,·n1 · .11H] /.O·cl1.1··wjd,· :-:11111·1·1•1·:, l11d: 'Jtllll)!. unly Pti.· ~·.11id1~ w .. \.11, wllil·l1 w.-H• placed JO 1·111 .1'1·11111 tl1t• :'1id.·wa..ll

oJ the .f Jn 1 nc.

3. 1. 2. 3 Wicl~' Source. The widest source used i11 the

c.xperi.n1ents occupied half the width of the 110-crn -wide fhune.

This source was used to study mixing of two wide parallel streams of the sa1ne depth and flowing with the same velocity but having

different concentrations or densities. This wide source, shown

in Fig. 3. 7, was made by partitioning the flume along part of its l1·11µ;th wit.h a longitudinal di vidi.ng wall and spraying a tracer into

<llh' :3lcl1· <Jf t.IJ<• flu1nc. nc·c-a111:1e of the 1:1y·nnnc•t:ry of t111~ apparat:ui:;.

on1• can observe sin1.ultaneously ir1 one experi.n1ent th.e rnixing of a heavy fluid into a light one, and a light fluid into a heavy one.

The dividing wall was 7. 3 meters long and 20 cm high. The

upstream end was 5. 3 meters downstream from the screen in the

inlet box of the flume. The wall was made from three pieces of 18-gage

(0. 049-inch) galvanized steel sheet which were put end to end. Each